Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model

Abstract

In this paper, an extension is paid to an idea of fractal and fractional derivatives which has been applied to a number of ordinary differential equations to model a system of partial differential equations. As a case study, the fractal fractional Schnakenberg system is formulated with the Caputo operator (in terms of the power law), the Caputo-Fabrizio operator (with exponential decay law) and the Atangana-Baleanu fractional derivative (based on the Mittag-Liffler law). We design some algorithms for the Schnakenberg model by using the newly proposed numerical methods. In such schemes, it worth mentioning that the classical cases are recovered whenever α=1 and β=1. Numerical results obtained for different fractal-order (β∈(0,1)) and fractional-order (α∈(0,1)) are also given to address any point and query that may arise.

Keywords

• 34A34
• 35A05
• 35K57
• 65L05
• 65M06
• 93C10